Verify the identity. \( \sin ^{2} \frac{\theta}{2}=\frac{\sec \theta-1}{2 \sec \theta} \) Use the appropriate power-reducing formula and rewrite the left side of the identity. \( \frac{1-\cos \theta}{2} \) (Simplify your answer.) Rewrite the expression from the previous step by multiplying the numerator and denominator by sec \( \theta \). Multiply and distribute in the numerator. \( \square \) (Do not simplify.)

To verify the identity, start with the left side \( \sin^{2} \frac{\theta}{2} \). Using the power-reducing formula, we have: \[ \sin^{2} \frac{\theta}{2} = \frac{1 - \cos \theta}{2}. \] Next, when you rewrite this expression and multiply both the numerator and denominator by \(\sec \theta\), it looks like this: \[ \frac{1 - \cos \theta}{2} \cdot \frac{\sec \theta}{\sec \theta} = \frac{(1 - \cos \theta) \sec \theta}{2 \sec \theta}. \] Now, distribute \(\sec \theta\) in the numerator: \[ 1 \cdot \sec \theta - \cos \theta \cdot \sec \theta = \sec \theta - 1. \] So we have: \[ \frac{\sec \theta - 1}{2 \sec \theta}. \] This confirms our original identity as true!